Open questions posed by Genoni and Paris in 'Quantifying non-Gaussianity for quantum information' (Phys. Rev. A 82, 052341, 2010) concern the two distance-type non-Gaussianity measures for continuous-variable states: the Hilbert–Schmidt measure δA\delta_A and the quantum-relative-entropy measure δB\delta_B, plus the induced operation-level measure δ[E]\delta[\mathcal{E}]. They are the general relation between δA\delta_A and δB\delta_B beyond the purity-weighted inequality δB≥δAμ[ϱ]\delta_B\ge\delta_A\mu[\varrho]; whether distillable entanglement admits a maximum at fixed non-Gaussianity of the resource state; a quantitative link between non-Gaussianity and quantum Fisher information when the parameter variation changes the covariance matrix; computable evaluation of δ[E]\delta[\mathcal{E}]; and the single-mode conjecture δA≤1/2\delta_A\le 1/2. None of these questions is known to be resolved in the literature; they remain open in their original generality.

Background

Continuous-variable (CV) quantum information encodes information in systems whose observables have continuous spectra, typically the quadrature operators (amplitude-like and phase-like linear combinations of the annihilation and creation operators of bosonic field modes, as in quantum optics). The workhorse states are the Gaussian states: states whose Wigner function — a quasi-probability distribution over phase space — is a Gaussian. A Gaussian state is completely specified by the first moments of the quadratures and by the covariance matrix (the table of second moments). Gaussian states and Gaussian operations (those generated by Hamiltonians at most quadratic in the field operators, which map Gaussian states to Gaussian states) are efficiently classically simulable, so the non-Gaussian character of states and operations is the resource behind genuinely quantum advantages in CV quantum information: it is necessary for entanglement distillation, it can enhance parameter estimation, and it is required for universal CV quantum computation.

To reason about non-Gaussianity as a resource one must quantify it. Genoni, Paris and Banaszek (2007) proposed measuring the non-Gaussianity of a state ϱ\varrho by its distinguishability from a reference Gaussian state τ\tau chosen to have exactly the same first and second moments as ϱ\varrho. With the Hilbert–Schmidt distance this yields the measure δA[ϱ]=DHS2[ϱ,τ]/μ[ϱ]\delta_A[\varrho]=D_{\rm HS}^2[\varrho,\tau]/\mu[\varrho], where DHS2[ϱ,τ]=Tr[(ϱ−τ)2]/2D_{\rm HS}^2[\varrho,\tau]={\rm Tr}[(\varrho-\tau)^2]/2 and μ[ϱ]=Tr[ϱ2]\mu[\varrho]={\rm Tr}[\varrho^2] is the purity; the normalization by purity makes the measure sensitive relative to the state's own 'size'. The same authors (2008) proposed the relative-entropy measure δB[ϱ]=S(ϱ∥τ)=S(τ)−S(ϱ)\delta_B[\varrho]=S(\varrho\Vert\tau)=S(\tau)-S(\varrho), the difference between the von Neumann entropy of the reference Gaussian and that of the state itself — a quantum analogue of the classical negentropy. Both measures vanish exactly on Gaussian states, are invariant under Gaussian unitary transformations, and δB\delta_B is non-increasing under Gaussian channels (noisy quantum channels that map Gaussian states to Gaussian states). A third, Wehrl-entropy-based measure (Ivan, Kumar and Simon) exists but is not even invariant under all Gaussian unitaries.

Genoni and Paris (2010) compared δA\delta_A and δB\delta_B on a 'zoology' of non-Gaussian states — Fock states (number eigenstates ∣n⟩|n\rangle), photon-added and photon-subtracted states (obtained by applying the photon creation or annihilation operator to a reference state such as a thermal or coherent state), superpositions of coherent states ('Schrödinger cat' states) — finding the same qualitative behaviour but provably different orderings: there exist pairs ϱ1,ϱ2\varrho_1,\varrho_2 with δA[ϱ1]>δA[ϱ2]\delta_A[\varrho_1]>\delta_A[\varrho_2] yet δB[ϱ1]<δB[ϱ2]\delta_B[\varrho_1]<\delta_B[\varrho_2]. The only general relation known is δB[ϱ]≥δA[ϱ] μ[ϱ]\delta_B[\varrho]\ge\delta_A[\varrho]\,\mu[\varrho], a consequence of the inequality between the quantum relative entropy and the squared Hilbert–Schmidt distance. The paper also: (a) lifted the state measures to operations, defining δ[E]=max⁡ϱ∈Gδ[E(ϱ)]\delta[\mathcal{E}]=\max_{\varrho\in{\cal G}}\delta[\mathcal{E}(\varrho)] over the set G{\cal G} of Gaussian input states — a quantity vanishing exactly on Gaussian maps (those sending every Gaussian input to a Gaussian output); (b) showed that δB\delta_B sharpens the known extremality of Gaussian states, at fixed covariance matrix, for communication quantities such as the Holevo bound, mutual information and conditional entropy; (c) connected δB\delta_B to parameter estimation: the quantum Fisher information H(λ)H(\lambda) — which through the quantum Cramér–Rao bound Var(λ)≥1/(MH(λ)){\rm Var}(\lambda)\ge 1/(M H(\lambda)) sets the best precision for estimating a parameter λ\lambda from MM measurements, and equals four times the squared Bures distance (a statistical distinguishability metric) between infinitesimally neighbouring states — is upper-bounded by δB[ϱλ+dλ]\delta_B[\varrho_{\lambda+d\lambda}] whenever the perturbed state shares the covariance matrix of the unperturbed Gaussian reference; and (d) analyzed entanglement distillation (concentrating the entanglement shared by two parties into fewer, more strongly entangled pairs by local operations and classical communication), where Gaussian-only protocols cannot distill entanglement at all, and found that the entanglement distilled by photon-subtraction protocols and the non-Gaussianity of the resource need not track each other.

Since 2018 the quantification of non-Gaussianity has additionally been reframed in the language of quantum resource theories — free operations being Gaussian operations (with classical feed-forward), and resource quantifiers being monotones that never increase under free operations — with computable monotones (Takagi and Zhuang; Albarelli, Genoni, Paris and Ferraro; Zhuang, Shor and Shapiro) and operational interpretations in terms of channel-discrimination advantage (Regula, Lami, Ferrari and Takagi). These frameworks complement rather than replace the δA/δB\delta_A/\delta_B program: they do not answer the questions the 2010 paper left open, which are the subject of this problem.

Problem Statement

Settle the open questions that Genoni and Paris (Phys. Rev. A 82, 052341 (2010)) left for their non-Gaussianity quantification framework. The five parts share one scientific objective — establishing whether the two distance-type measures δA\delta_A (Hilbert–Schmidt) and δB\delta_B (quantum relative entropy), together with the induced operation-level measure δ[E]\delta[\mathcal{E}], constitute a mathematically complete and operationally meaningful quantification of non-Gaussianity for continuous-variable states and operations. The five parts are independent deliverables: each may be settled on its own, and resolving any one of them counts as a solution regardless of whether the others are resolved. Part 1 contains two independent questions, (1a) and (1b):

  1. Measure relations. Two independent questions:

    • (1a) Apart from the purity-weighted inequality δB[ϱ]≥δA[ϱ]μ[ϱ]\delta_B[\varrho]\ge\delta_A[\varrho]\mu[\varrho], determine what general relations hold between δA\delta_A and δB\delta_B — order relations, monotonicity implications, or class-restricted versions valid for restricted families of states (e.g. pure, Fock-diagonal, or single-mode states) — or prove that no stronger relation exists.
    • (1b) Determine whether different non-Gaussianity measures carry genuinely different operational meanings — whether an operational task (for example a state-discrimination or parameter-estimation task) exists whose performance ordering of states or operations disagrees with the ordering induced by the measure, and in particular whether δA\delta_A admits an operational meaning beyond its connection with the phase-space distance. The paper conjectures, by analogy with entanglement measures, that there is no unique non-Gaussianity measure and that different measures correspond to different operational meanings.

  2. Non-Gaussianity and entanglement distillation. Determine whether there exists a maximum amount of distillable entanglement at fixed non-Gaussianity of the resource state: for a fixed value of δA\delta_A (or δB\delta_B) of the input state, is the entanglement distillable by any protocol — Gaussian or non-Gaussian — bounded above, and if so what is the bound?

  3. Non-Gaussianity and parameter estimation. Establish, or rule out, a general quantitative connection between non-Gaussianity and the quantum Fisher information when the parameter variation also changes the covariance matrix, generalizing the fixed-covariance theorem: if τλ\tau_\lambda is Gaussian and an infinitesimal change of λ\lambda drives it into a state ϱλ+dλ\varrho_{\lambda+d\lambda} with the same covariance matrix, then δB[ϱλ+dλ]\delta_B[\varrho_{\lambda+d\lambda}] upper-bounds H(λ)H(\lambda); no general result exists when the covariance matrix varies.

  4. Computation of the operation-level measure. Develop rigorous methods to evaluate δ[E]=max⁡ϱ∈Gδ[E(ϱ)]\delta[\mathcal{E}]=\max_{\varrho\in{\cal G}}\delta[\mathcal{E}(\varrho)] for a general quantum operation E\mathcal{E} (with either the δA\delta_A or δB\delta_B state measure), for which the maximization over the Gaussian inputs is in general intractable.

  5. Single-mode bound (Conjecture A5 of the source). Prove or refute: for every state ϱ\varrho of a single bosonic mode, δA[ϱ]≤1/2\delta_A[\varrho]\le 1/2. Numerical evidence (random states truncated in the Fock basis, Fock states and their superpositions, cat states) supports the bound, which is approached asymptotically by Fock states ∣n⟩|n\rangle as n→∞n\to\infty according to the closed-form evaluation in the source, but no proof or counterexample is known.

References

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